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Simplifying 0.5b(b + 3) = 27 Reorder the terms: 0.5b(3 + b) = 27 (3 * 0.5b + b * 0.5b) = 27 (1.5b + 0.5b2) = 27 Solving 1.5b + 0.5b2 = 27 Solving for variable 'b'. Reorder the terms: -27 + 1.5b + 0.5b2 = 27 + -27 Combine like terms: 27 + -27 = 0 -27 + 1.5b + 0.5b2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -54 + 3b + b2 = 0 Move the constant term to the right: Add '54' to each side of the equation. -54 + 3b + 54 + b2 = 0 + 54 Reorder the terms: -54 + 54 + 3b + b2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + 3b + b2 = 0 + 54 3b + b2 = 0 + 54 Combine like terms: 0 + 54 = 54 3b + b2 = 54 The b term is 3b. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3b + 2.25 + b2 = 54 + 2.25 Reorder the terms: 2.25 + 3b + b2 = 54 + 2.25 Combine like terms: 54 + 2.25 = 56.25 2.25 + 3b + b2 = 56.25 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 56.25 Calculate the square root of the right side: 7.5 Break this problem into two subproblems by setting (b + 1.5) equal to 7.5 and -7.5.Subproblem 1
b + 1.5 = 7.5 Simplifying b + 1.5 = 7.5 Reorder the terms: 1.5 + b = 7.5 Solving 1.5 + b = 7.5 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = 7.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 7.5 + -1.5 b = 7.5 + -1.5 Combine like terms: 7.5 + -1.5 = 6 b = 6 Simplifying b = 6Subproblem 2
b + 1.5 = -7.5 Simplifying b + 1.5 = -7.5 Reorder the terms: 1.5 + b = -7.5 Solving 1.5 + b = -7.5 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = -7.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -7.5 + -1.5 b = -7.5 + -1.5 Combine like terms: -7.5 + -1.5 = -9 b = -9 Simplifying b = -9Solution
The solution to the problem is based on the solutions from the subproblems. b = {6, -9}
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